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The annihilating-ideal graph of commutative rings. I. (English) Zbl 1276.13002

Summary: Let \(R\) be a commutative ring, with \(\mathbb{A}(R)\) its set of ideals with nonzero annihilator. In this paper and its sequel [ibid. 10, No. 4, 741–753 (2011; Zbl 1276.13003)], we introduce and investigate the annihilating-ideal graph of \(R\), denoted by \(\mathbb{A}\mathbb{G}(R)\). It is the (undirected) graph with vertices \(\mathbb{A}(R)^*= \mathbb{A}(R)\setminus\{(0)\}\), and two distinct vertices \(I\) and \(J\) are adjacent if and only if \(IJ = (0)\). First, we study some finiteness conditions of \(\mathbb{A}\mathbb{G}(R)\). For instance, it is shown that if \(R\) is not a domain, then \(\mathbb{A}\mathbb{G}(R)\) has ascending chain condition (respectively, descending chain condition) on vertices if and only if \(R\) is Noetherian (respectively, Artinian). Moreover, the set of vertices of \(\mathbb{A}\mathbb{G}(R)\) and the set of nonzero proper ideals of \(R\) have the same cardinality when \(R\) is either an Artinian or a decomposable ring. This yields for a ring \(R\), \(\mathbb{A}\mathbb{G}(R)\) has \(n\) vertices (\(n\geq 1\)) if and only if \(R\) has only \(n\) nonzero proper ideals. Next, we study the connectivity of \(\mathbb{A}\mathbb{G}(R)\). It is shown that \(\mathbb{A}\mathbb{G}(R)\) is a connected graph and \(\mathrm{diam}\mathbb{A}\mathbb{G}(R) \leq 3\) and if \(\mathbb{A}\mathbb{G}(R)\) contains a cycle, then \(\mathrm{gr}\mathbb{A}\mathbb{G}(R) \leq 4\). Also, rings \(R\) for which the graph \(\mathbb{A}\mathbb{G}(R)\) is complete or star, are characterized, as well as rings \(R\) for which every vertex of \(\mathbb{A}\mathbb{G}(R)\) is a prime (or maximal) ideal. In part II we shall study the diameter and coloring of annihilating-ideal graphs.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
05C75 Structural characterization of families of graphs

Citations:

Zbl 1276.13003
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References:

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